## matrix – Manipulation list: {{{a, b}, {c, d}}, {{e, f}, {gh}}} in {{a, b}, {c, d}, {e, f}, {g, h}}

Hello, I would like to transform the matrix:

``````{{{184.586, 1.02758 * 10 ^ 8}, {139.94, 4.07249 * 10 ^ 7}, {117.72,
2.26123 * 10 ^ 7}, {109.528, 1.51412 * 10 ^ 7}, {95.7636,
8.68593 * 10 ^ 6}, {78.125, 5.16118 * 10 ^ 6}, {66.6777,
3.1014 * 10 ^ 6}}, {{183.505, 1.01147 * 10 ^ 8}, {138.25,
4.02021 * 10 ^ 7}, {115.814, 2.24641 * 10 ^ 7}, {108.22,
1.50985 * 10 ^ 7}, {91.3082, 9.28241 * 10 ^ 6}, {78.1087,
6.03119 * 10 ^ 6}, {65.9855, 3.56718 * 10 ^ 6}}}
``````

with the same transformation that

``````{{{a, b}, {c, d}}, {{e, f}, {gh}}}
``````

inside

``````{{a, b}, {c, d}, {e, f}, {g, h}}
``````

How can I do please?

hg

## rt.representation theory – Expressing \$ sum_ {g in [G/H]} ge_Hg ^ {- 1} in Z ( mathbb {C}[G]) \$ in terms of primitive central idempotentes?

Suppose $$G$$ it's a finite group, and $$H$$ a subgroup For an irreducible character. $$chi$$ of $$G$$, there is a central idempotent in group algebra. $$mathbb {C}[G]$$:
$$e_ chi = frac { chi (1)} {| G |} sum_ {g in G} chi (g ^ {- 1}) g.$$

I write $$e_H: = e_ {1_H} = frac {1} {| H |} sum_ {h in H} h$$ for the idempotent in $$mathbb {C}[H]$$ corresponding to the trivial character of $$H$$. Yes $$[G/H]$$ Denotes a complete set of left coset representatives from $$H$$ in $$G$$, then by construction the element
$$sum_ {g in [G/H]} ge_Hg ^ {- 1}$$
it is central in $$mathbb {C}[G]$$. Is there any way to explicitly extract which characters have their corresponding central idempotent in this linear combination and / or its multiplicity? Just looking at Mackey's formula and since then $$ge_Hg ^ {- 1} = e_ {gHg ^ {- 1}}$$, my guess is that they can be characters that are constituents of $$operatorname {Ind} ^ G_H (1_H)$$, or something similar, but I'm not sure.

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